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TO DETERMINE THE DERIVATIVE OF F OF X
= SQUARE ROOT OF X CUBED - 4,
WE NEED TO RECOGNIZE THIS AS A COMPOSITE FUNCTION
WHERE THERE IS AN INNER FUNCTION AND AN OUTER FUNCTION,
AND THEREFORE WE WILL HAVE TO APPLY THE CHAIN RULE.
AND THE CHAIN RULE TELLS US TO DETERMINE THE DERIVATIVE
OF A COMPOSITE FUNCTION
WE NEED TO DETERMINE THE DERIVATIVE OF THE OUTER FUNCTION
AND THEN MULTIPLY IT BY THE DERIVATIVE
OF THE INNER FUNCTION.
SO WHEN WE HAVE A COMPOSITE FUNCTION
WE WANT TO LET U EQUAL THE INNER FUNCTION.
SO IN THIS CASE U WOULD BE X TO THE 3rd - 4.
AND LET'S GO AHEAD AND WRITE THAT OUT OVER HERE.
U = X CUBED - 4,
WHICH MEANS WE COULD WRITE THIS AS THE SQUARE ROOT OF U
AND WE KNOW BY NOW IF WE HAVE A SQUARE ROOT
WE WANT TO REWRITE THIS USING A RATIONAL EXPONENT
WHERE THE EXPONENT HERE IS ONE AND THE INDEX IS TWO.
SO THIS IS EQUAL TO U TO THE 1/2 POWER.
ONCE WE HAVE OUR FUNCTION WRITTEN IN TERMS OF U
IT'S A VERY STRAIGHT FORWARD PROCESS TO APPLY THE CHAIN RULE.
WE NEED TO FIND THE DERIVATIVE OF U TO THE 1/2
AS WE NORMALLY DO
AND THEN MULTIPLY IT BY THE DERIVATIVE OF U.
AND USUALLY ONCE YOU LEARN THE CHAIN RULE
ALL OF YOUR DIFFERENTIATION RULES ARE REWRITTEN
SO THAT THEY CONTAIN A U
AS WE SEE HERE FOR THE EXTENDED OR GENERAL POWER RULE.
AND NOTICE HOW THE ONLY DIFFERENCE HERE
IS THAT WE HAVE A U HERE INSTEAD OF X,
AND THEN WE HAVE THE DERIVATIVE IN TERMS OF U x U PRIME,
WHICH AGAIN IS JUST THE CHAIN RULE
TELLING US TO FIND THE DERIVATIVE OF THE OUTER FUNCTION
AND THEN MULTIPLY IT BY THE DERIVATIVE
OF THE INNER FUNCTION.
SO F PRIME OF X IS GOING TO BE EQUAL TO THE DERIVATIVE OF U
TO THE 1/2.
THAT WOULD BE 1/2 U TO THE 1/2 - 1 x U PRIME.
AND NOW WE NEED TO REWRITE THIS DERIVATIVE IN TERMS OF X
RATHER THAN U.
SO WE'LL REPLACE U WITH X TO THE 3rd -4
AND WE'LL REPLACE U PRIME WITH THE DERIVATIVE OF U
WHICH WOULD BE 3X SQUARED.
SO WE'RE GOING TO HAVE 1/2 x U, WHICH IS X TO THE 3RD - 4,
TO THE POWER OF 1/2 - 1.
THAT'S -1/2 x U PRIME AND U PRIME IS 3X SQUARED.
SO THIS IS OUR DERIVATIVE FUNCTION,
BUT NOW WE DO HAVE TO SIMPLIFY THIS.
SO IF WE THINK OF THIS AS BEING OVER ONE WE CAN MORE THIS DOWN
SO THAT OUR EXPONENT WOULD BE POSITIVE.
LET'S ALSO PUT THIS OVER ONE
SO WE KNOW WHAT PART IS IN THE NUMERATOR
AND WHAT'S IN THE DENOMINATOR.
SO OUR DERIVATIVE FUNCTION IS GOING TO BE EQUAL
TO THE FRACTION
WHERE THE NUMERATOR WOULD BE 3X SQUARED
AND THE DENOMINATOR WOULD BE 2
x (X TO THE 3rd - 4 TO THE POSITIVE 1/2 POWER).
AND IF WE WANTED TO WE COULD WRITE THIS
AS THE SQUARE ROOT OF X CUBED -4.
SO LET'S GO AHEAD AND DO THAT.
WE HAVE 3X SQUARED,
AND OUR DENOMINATOR IS 2 x (SQUARE ROOT OF X CUBED - 4).
SO AS YOU CAN SEE,
AS LONG AS WE'VE IDENTIFIED THE INNER FUNCTION AS U,
WE WRITE THE FUNCTION IN TERMS OF U,
APPLYING THE CHAIN RULE IS PRETTY STRAIGHT FORWARD.
WE FIND THE DERIVATIVE OF U AND THEN MULTIPLY IT BY U PRIME.
I HOPE YOU HAVE FOUND THIS HELPFUL.